Question 99
Let $f, g$ be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is $\mathbf{x}_0 = (0,0)$, and for every $n \ge 1$, his jump is
$$
\mathbf{x}_n - \mathbf{x}_{n-1} = (\pm f(n), \pm g(n)) \text{ or } (\pm g(n), \pm f(n)),
$$
with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location $(0,0)$ infinitely many times when
(a) $f, g$ are polynomials with integer coefficients?
(b) $f, g$ are any pair of functions from the positive integers to the integers?
$$
\mathbf{x}_n - \mathbf{x}_{n-1} = (\pm f(n), \pm g(n)) \text{ or } (\pm g(n), \pm f(n)),
$$
with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location $(0,0)$ infinitely many times when
(a) $f, g$ are polynomials with integer coefficients?
(b) $f, g$ are any pair of functions from the positive integers to the integers?